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DOI: 10.4230/LIPIcs.MFCS.2017.5
URN: urn:nbn:de:0030-drops-80985
URL: http://drops.dagstuhl.de/opus/volltexte/2017/8098/
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Durand, Bruno ; Romashchenko, Andrei

On the Expressive Power of Quasiperiodic SFT

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LIPIcs-MFCS-2017-5.pdf (0.5 MB)


Abstract

In this paper we study the shifts, which are the shift-invariant and topologically closed sets of configurations over a finite alphabet in Z^d. The minimal shifts are those shifts in which all configurations contain exactly the same patterns. Two classes of shifts play a prominent role in symbolic dynamics, in language theory and in the theory of computability: the shifts of finite type (obtained by forbidding a finite number of finite patterns) and the effective shifts (obtained by forbidding a computably enumerable set of finite patterns). We prove that every effective minimal shift can be represented as a factor of a projective subdynamics on a minimal shift of finite type in a bigger (by 1) dimension. This result transfers to the class of minimal shifts a theorem by M.Hochman known for the class of all effective shifts and thus answers an open question by E. Jeandel. We prove a similar result for quasiperiodic shifts and also show that there exists a quasiperiodic shift of finite type for which Kolmogorov complexity of all patterns of size n\times n is \Omega(n).

BibTeX - Entry

@InProceedings{durand_et_al:LIPIcs:2017:8098,
  author =	{Bruno Durand and Andrei Romashchenko},
  title =	{{On the Expressive Power of Quasiperiodic SFT}},
  booktitle =	{42nd International Symposium on Mathematical Foundations of Computer Science (MFCS 2017)},
  pages =	{5:1--5:14},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-046-0},
  ISSN =	{1868-8969},
  year =	{2017},
  volume =	{83},
  editor =	{Kim G. Larsen and Hans L. Bodlaender and Jean-Francois Raskin},
  publisher =	{Schloss Dagstuhl--Leibniz-Zentrum fuer Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{http://drops.dagstuhl.de/opus/volltexte/2017/8098},
  URN =		{urn:nbn:de:0030-drops-80985},
  doi =		{10.4230/LIPIcs.MFCS.2017.5},
  annote =	{Keywords: minimal SFT, tilings, quasiperiodicityIn this paper we study the shifts, which are the shift-invariant and topologically closed sets of configurations}
}

Keywords: minimal SFT, tilings, quasiperiodicityIn this paper we study the shifts, which are the shift-invariant and topologically closed sets of configurations
Seminar: 42nd International Symposium on Mathematical Foundations of Computer Science (MFCS 2017)
Issue Date: 2017
Date of publication: 22.11.2017


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